How do you say "forum" in greek? I have been thinking about modern terms, and how to say them, but other than the vague αγορά, I can't think of anything else for "forum".
For keyboard I'd use the modern word, πληκτρολόγιο, which is good enough for the job. For an audio recording it would be ηχογράφηση, but what about "video"?

I generally don't talk about the internet etc (or frankly use Anc Greek in this way). You've got a few choices, you can follow the modern method and just use "forum" or calque it as agora as you say. For thread you can use θέμα following modern usage or calque again and use κλωστή or something. Δημοσίευση for posting and so on and forth.

I think if you're going to use anc Greek for modern things you've two options: Go entirely second sophistic and come up with new terms or borrow from the modern language. I don't necessarily think the latter option is more "authentic" or anything since a lot of technical vocabulary comes from clumsy katherevousa masturbation. So you're getting these words from a modern filter anyway even if they look ancient. But none of this is authentic, the ancients didn't have internets and whatnot. It has the advantage, however, of being used my modern, native, speakers and being easy to apprehend. I see nothing wrong in just using words like βιντεο and happily do so.

It depends on why you are using Ancient Greek to discuss modern things.

On our weather thread we have faced the same problem because even though they had weather (obviously) they didn't really think about it the way we do. On the basis that our aim is to learn Greek we either go for calques or extend the meaning of an ancient Greek word to cover something that at least reminds us of the actual ancient Greek meaning. Hence we use κλῖμαξ (ladder) for scale which means should we read of soldiers attacking a city using κλῖμαξ we will not be phased even though we are using the word in a way that Ancient Greeks never did.

Just making forum into a Greek word (φορος?) is no less authentic but doesn't server our purpose.